Extended Convergence of the Extremal Process of Branching Brownian Motion

2014/12/18 by Anton Bovier, Bovier, Anton, Lisa Hartung +1
Economics, Econometrics and Finance · Mathematics · #60G70 #60J80 #82B44 #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60G70 #msc:60J80 #msc:82B44

paper · pdf · doi:10.48550/arxiv.1412.5975

22 pages, 1 figure, revised version

openalex publication_date 2014/12/18 · arxiv created 2016/09/21 · arxiv updated 2016/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We extend the results of Arguin et al and A"ıdékon et al on the convergence of the extremal process of branching Brownian motion by adding an extra dimension that encodes the "location" of the particle in the underlying Galton-Watson tree. We show that the limit is a cluster point process on ℝ₊× ℝ where each cluster is the atom of a Poisson point process on ℝ₊× ℝ with a random intensity measure Z(dz) × Ce^{-√ 2x}dx, where the random measure is explicitly constructed from the derivative martingale. This work is motivated by an analogous result for the Gaussian free field by Biskup and Louidor.

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• The extremal point process of branching Brownian motion in ℝ^d
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• An ergodic theorem for the extremal process of branching Brownian motion